A note on the diophantine equation ${k\choose 2}-1=q^n+1$

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Department of Mathematics Zhanjiang Teachers College 524048 Zhanjiang, Guangdong P.R. China

In this note we prove that the equation

{k c h \infty s e 2} - 1 = q^{n} + 1

,

q \geq 2, n \geq 3

, has only finitely many positive integer solutions

(k, q, n)

. Moreover, all solutions

(k, q, n)

satisfy

k < 10^{10^{182}}

,

q < 10^{10^{165}}

and

n < 2 \cdot 10^{17}

.

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